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Consider the classic Vehicle Routing Problem.

In this paper, the author shows how to optimally partition an optimal TSP tour, into feasible routes. In other words he describes a "route-first cluster-second" approach for the problem, where the clustering part is optimal.

On page $2$, it is written:

Note that it is easily shown that an optimal TSP tour followed by an optimal set of vehicle routes does not necessarily lead to an optimal set of vehicle routes.

I am trying to come up with a very simple example where this is true. Can anyone help?

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Let out graph have $6$ vertices on a plane with Manhattan metric (for simplicity). Vertices are point with following coordinates: $v_1(0, 0), v_2(0, 1), v_3(5, 0), v_4(5, 1), v_5(10, 0), v_6(10, 1)$. Then $v_1v_2v_4v_6v_5v_3v_1$ is the optimal TSP tour. Let $v_1$ be depot and vehicle capacity equals 2. If you partition tour optimally you'll have to visit $v_5$ and $v_6$ by one vechicle (22 units length), $v_4$ and $v_2$ by anothere one (12 units length) and $v_3$ by the third one (10 units length), that is 44 units length in total. Optimal set of vehicle routes is the following: visit $v_5$ and $v_6$ by one vechicle (22 units length), $v_3$ and $v_4$ by another one (12 units length) and $v_2$ by the third one (2 units length), that is 36 units length in total.

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    How do you know that you have the optimal partition? – 2017-02-08
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    There are only two partitions possible. The other one has the same total length. – 2017-02-08
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    Given the tour $v_1-v_2-v_4-v_6-v_5-v_3-v_1$, I believe there are more than $2$ feasible partitions, for example the following partition is also feasible: $v_1-v_2-v_1$, $v_1-v_4-v_1$, $v_1-v_6-v_1$, $v_1-v_5-v_1$, $v_1-v_3-v_1$ But I think you are right...by inspection it is not difficult to see that the one you have proposed is the best one. – 2017-02-08